J. Peraire, S. Widnall
16.07 Dynamics
Fall 2008
Version 2.0
Lecture L14 - Variable Mass Systems: The Rocket Equation
In this lecture, we consider the problem in which the mass of the body changes during the motion, that is,
m is a function of t, i.e. m(t). Although there are many cases for which this particular model is applicable,
one of obvious importance to us are rockets. We shall see that a significant fraction of the mass of a rocket
is the fuel, which is expelled during flight at a high velocity and thus, provides the propulsive force for the
rocket.
As a simple model for this process, consider the cases sketched in a) through f) of the figure. In a) we
consider 2 children standing on a stationary flat car. At t = 0, they both jump off with velocity relative to
the flat car of u. (While this may not be a good model for children jumping off a flatcar, it is a good model
for the expulsion of mass from a rocket where the mass flow from the choked nozzle occurs at a relative
velocity u from the moving rocket.) In b) we consider that they jump off in sequence, each jumping with a
velocity u relative to the then velocity of the flatcar, which will be different for the 2nd jumper since the car
has begun to move as a result of jumper 1. For a), we have a final velocity of the flat car as
2mu
M + 2m
(1)
mu
mu
+
M + 2m M + m
(2)
V2R =
for b) we have
V22 =
where M is the mass of the flatcar, and m is the mass of the jumper. We can see the case b) gives a higher
final velocity. We now consider cases with more blocks; we introduce the notation VN N to be the final
velocity with N blocks coming off one at a time. We use the notation VN R to denote the reference velocity
if the N blocks come off together. If we consider 3 blocks, we can see by inspection that
V 3R =
3mu
M + 3m
(3)
and
mu
mu
mu
+
+
M + 3m M + 2m M + m
(4)
4mu
M + 4m
(5)
mu
mu
mu
mu
+
+
+
M + 4m M + 3m M + 2m M + m
(6)
V33 =
While for 4 blocks
V4R =
and
V44 =
1
Generalizing this to arbitrary N, we can write
VN R =
and
VN N =
N mu
M + Nm
N
�
i=1
mu
M + im
(7)
(8)
The figure shows the results for V100,100 /V100R , the increase in velocity as each block comes off. We choose
100 blocks with a mass m equal to .01 M; therefore the 100 blocks with m = .01 equal the mass M, the
propellant ratio of the ”rocket” is 50%. The final velocity of the cart is 40% higher than would be obtained
if the masses come off together. Momentum is conserved. The sum of the momentum of the cart plus all the
individual mass points is equal to zero. However, the momentum of all the particles left behind is of very
little interest to us. We care only about the final velocity of the cart.
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To analyze this question we must consider a system of variable mass, and the process by which it gains
velocity as a result of ejecting mass. While our previous discussion provided a discrete model of this process,
we now consider a continuous process, obtaining a relation for v(t) as a function of the momentum interaction
between the system and the external world.
We shall start by considering a body with velocity v and external forces F , gaining mass at a rate ṁ = dm/dt.
Let us look at the process of gaining a small amount of mass dm. Let v � be the velocity of dm before it is
captured by m, and let f represent the average value of the impulsive forces that dm exerts on m during
the short interval dt, in which the capturing takes place. By Newton’s third law, dm will experience a force
−f , exerted by m, over the same dt.
We can now examine the capture process from the point of view of dm and equate the impulse, −f dt, to
the change in linear momentum of dm,
−f dt = dm(v + dv − v � ).
(9)
Here, v + dv is the velocity of m (and dm) after impact. Analogously, from the point of view of m, we write
F dt + f dt = m(v + dv) − mv = mdv.
(10)
The term dm dv in equation (9) is a higher order term and will disappear when we take limits. The impulse
due to the contact force can be eliminated by combining equations (9) and (10),
F dt − dm(v − v � ) = mdv,
or, dividing through by dt,
m
dv
dm
dm
= F − (v − v � )
= F + (v � − v)
.
dt
dt
dt
(11)
Here, v � − v is the velocity of dm relative to m. This expression is valid when dm/dt > 0 (mass gain) and
when dm/dt < 0 (mass loss). If we compare this expression to the more familiar form of Newton’s law for
�
a particle of fixed mass m dv
dt = F , we see that the term (v − v)dm/dt is an additional force on m which is
due to the gain (or loss) of mass. Equation (11) can also be written as
d(mv)
dm
= F + v�
,
dt
dt
3
where v � is the velocity of the captured (or expelled) mass relative to the velocity of the mass m. This shows
that, for systems involving variable mass, the usual expression stating conservation of linear momentum,
d(mv)/dt = F , is only applicable when the initial (final) velocity of the captured (expelled) mass, v � , is
zero. The behavior of m(t) is not an unknown, but is specified according to the characteristics of the rocket.
In most cases,
dm
dt
is a constant and negative. In some cases, the behavior of m(t) may be determined by a
control system. In any case, it is a given quantity.
The Rocket Equation
We consider a rocket of mass m, moving at velocity v and subject to external forces F (typically gravity
and drag). The rocket mass changes at a rate ṁ = dm/dt, with a velocity vector c relative to the rocket.
We shall assume that the magnitude of c is constant.
The velocity of the gas observed from a stationary frame will be v � = v + c. In this frame, c is a vector
aligned along the flight path in a negative direction. c = −ct, where t is the unit direction along the flight
path. Thus,
m
dv
dm
=F +c
.
dt
dt
(12)
The term T = c dm/dt is called the thrust of the rocket and can be interpreted as an additional force on the
rocket due to the gas expulsion.
Equation (12) is a vector equation which can be projected along the direction of v (tangent to the path).
Thus,
m
dv
dm
= Ft − c
= Ft + T,
dt
dt
(13)
where Ft is the tangential component of F , v and c are the magnitudes of v and c respectively, and we have
assumed that c is parallel and has opposite direction to v. The magnitude of the thrust is T = −cṁ. Note
that for a rocket, ṁ will be negative (mass is lost). If the force Ft is known, this equation can be integrated
in time to yield an expression for the velocity as a function of time.
Let us consider some simple cases:
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No External Forces: Ft = 0
If gravity and drag effects are neglected, we have,
m
dv
dm
= −c
dt
dt
or, integrating between an initial time t0 , and a final time t,
Δv = v − v0 = −c(ln mf − ln m0 ) = −c ln
m
m0
= c ln
.
mf
m0
(14)
Alternatively, this expression can be cast as the well known rocket equation,
m = m0 e−Δv/c ,
(15)
which gives the mass of the rocket at a time t, as a function of the initial mass m0 , Δv, and c. The mass of
the propellant, mpropellant , is given by,
mpropellant = m0 − m = m0 (1 − e−Δv/c ).
From the above equations, we see that for a given Δv and m0 , increasing c increases m (payload plus
structure) and decreases mpropellant . Unfortunately, we can only choose c as high as the current technology
will allow. For current chemical rockets, c ranges from 2500-4500 m/sec. Ion engines can have c’s of roughly
105 m/sec. However, since relatively few particles are involved, the thrust is quite low, whereas chemical
rockets have high mass flows.
Equation (13) has some quite remarkable properties, as sketched below. First, the ΔV for a given m0 /mf is
independent of time. More strikingly, if we compare the final ΔV to the flat car situation where all the mass
was ejected instantaneously rather than continuously, we see a substantial gain in ΔV from the continuous
process. However, the final ΔV for the continuous process is independent of the actual time span of the
process, as shown in the sketch. Thus in the limit as the continuous process occurs quickly, we will obtain
the result for the continuous process. We will refer to the rapid continuous process as a ΔV impulse. When
we consider rocket thrusting during orbital maneuvers, we will consider only impulsive thrust.
A second property, is that the final ΔV from a series of firings is independent of the details and a function only
0
of the total mass change over the processes. Consider a first thrust producing a ΔV . Then ΔV1 = c ln m
m1 .
1
Now the mass is m1 and a second thrust produces a ΔV2 = c ln m
m2 . By a property of logarithms, adding
m1
m0
0
these two ΔV � s together gives ΔVf = ΔV1 + ΔV2 = c ln m
m1 + c ln m2 = c ln m2 where m2 is the final mass
of the vehicle mf .
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Another property is that the relationship between thrust and ΔV is a positive scaler, always additive.
Whether we thrust in the positive or negative direction,–or to the side– it always consumes propellant, we
never get anything back
˙ and substitute into equation 14 to obtain an
Note that if m
˙ is constant, we can write m(t) = m0 + mt,
expression for v as a function of t,
v = v0 − c ln(1 +
ṁ
t)
m0
.
Recall that according to the convention used, ṁ is negative as the mass decreases with time.
Gravity: Ft = −mg
A constant gravitational field acting in the opposite direction to the velocity vector can be easily incorporated.
In this case, equation 13 becomes,
m
dv
dm
= −mg − c
,
dt
dt
which can be integrated to give
v = v0 − c ln
m
m
m − m0
− gt = v0 − c ln
−g
.
m0
m0
ṁ
(16)
This solution assumes that cṁ > m0 g at t=0. If this is not true, the rocket will sit on the pad, burning fuel
until the remaining mass satisfies this requirement.
Gravity Plus Drag: Ft = −mg − D
The effect of the drag force, D, is harder to quantify. It turns out that for many important applications drag
effects are very small. The drag force is characterized in terms of a drag coefficient, CD . Thus,
D=
1 2
ρv ACD ,
2
6
where A is the cross-sectional area of the rocket. The air density changes with altitude z, and may be
approximated by
ρ = ρ0 e−z/H ,
where H ≈ 8000 m is the so-called “scale height” of the atmosphere, and ρ0 is the air density at sea level.
It turns out that the differential equation that results for the velocity cannot be integrated explicitly and,
in practice, needs to be integrated numerically. It is interesting to note, however, that the effect of drag
losses is usually quite small, and it is often reasonable to ignore it in a first calculation. In order to see the
importance of D versus the effect of gravity, we can estimate the value of the ratio D/mg. At conditions
typical for maximum drag, ρ ≈ 0.25 kg/m3 and v = 700 m/s. Considering a rocket of 12, 000 kg with a cross
section of A = 1 m2 and CD = 0.2, we have,
ρACD v 2
0.25 × 1 × 0.2 × 7002
=
= 0.021 ,
2mg
2 × 12, 000 × 9.8
which indicates that the drag force is only about 2% of the gravity force.
Note (Optional)
Gravity Losses [1]
Let us consider a rocket providing a constant thrust T = −cṁ which is launched vertically upwards from
rest. Neglecting drag but considering gravity forces, the velocity is given by expression (16) which can be
re-written as
�
�
1
1−µ
v = c ln −
,
µ
n
(17)
where we have introduced µ = m/m0 and the thrust induced acceleration measured in “gees”, n =
−cm/(gm
˙
0 ). Now, v = dz/dt, and the above equation can be integrated to give
�
�
c2
1
(1 − µ)2
z=
1 − µ ln − µ −
.
gn
µ
2n
(18)
If we consider equations (17) and (18) at burnout time (when all the propellant is consumed), the mass
fraction will be µ = (m0 − mpropellant )/m0 . After this time, the rocket will coast in free-flight. The total
energy per unit mass at this (and all subsequent times) is
� �
�
�2
v2
1
1
1 − µ − ln(1/µ)
2
E=
+ gz = c
ln
+
.
2
2
µ
n
Since, 0 < µ < 1, 1 − µ − ln(1/µ) is negative and consequently, higher n means larger final energy per unit
mass E, tending to the ideal limit
Eideal =
1
2
�
�2
1
c ln
,
µ
which is that obtained for an impulsive start. Reducing n means applying smaller thrust for a longer time,
and, as the above equation shows there is a price to be paid in the eventual energy of the payload. One way
to quantify this loss is to define an equivalent impulsive velocity (v0 )eq such as to achieve the energy per
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unit mass E in a very short time, hence, at ground level,
��
�2
1
1 − µ − ln(1/µ)
(v0 )eq = c
ln
+2
,
µ
n
and to compare this to the ideal impulsive velocity
(v0 )ideal = c ln
1
.
µ
Some representative fractional losses ((v0 )ideal − (v0 )eq )/(v0 )ideal , are shown below,
µ = 0.7
µ = 0.5
µ = 0.3
n = 1.5
0.3628
0.3188
0.2676
n=3
0.1615
0.1444
0.1235
n = 10
0.0456
0.0410
0.0354
The table shows losses of ∼ 15% for n = 3, ∼ 4% for n = 10. (Recall that n must be greater than 1 for
immediate lift off from the launch pad.) The losses are severe for n as low as 1.5. The figure shows a graph
of these losses as a function on n for µ = .8, .7, .5, and .3.
Example
Single vs. Two Stage Rockets [1]
Single stage
To achieve an orbital speed of v = 7600 m/s, we require an ideal Δv of about 9000 m/s where the extra
velocity is needed to overcome gravity and drag. Chemical rockets produce exhaust jets at velocities of
c ≈ 2500 − 4500 m/s. Using the higher c, if we wish to place a payload in orbit with a single stage rocket,
we have a mass ratio (mass at burn out, mf , divided by initial mass, m0 ) of
mf
= e−9000/4500 = e−2 = 0.135 .
m0
8
The mass mf must include all components of the rocket infrastructure, including the engine, empty tank,
guidance equipment, etc., as well as the payload. The mass of the propellant will be m0 − mf = (1 −
0.135)m0 = 0.865m0 . If we assume that the tank plus engine are the main contributors to the weight of the
rocket and weigh 10% of the propellant, 0.087m0 , we have,
m1stage
payload = (0.135 − 0.087)m0 = 0.048m0 .
Clearly, there is not much margin here, and, in fact, single-stage-to-orbit vehicles have yet to be engineered
successfully. The alternative solution is to subdivide the rocket, so that empty tanks are dropped when they
are no longer needed.
Two stages
Consider now a two stage vehicle. We shall again assume that each empty tank plus its engine weighs 10%
of the propellant it carries. The required Δv is now subdivided into two Δv’s of 4500 m/s each.
First stage
If m0 is the initial mass and m1 is the mass after burn out, we have,
m1 = e−4500/4500 m0 = e−1 m0 = 0.368m0
The fuel burnt will be m0 − m1 = (1 − 0.368)m0 = 0.632m0 , and the weight of the tank and engine to be
dropped will be 0.0632m0 , leaving an initial mass for the second stage, m2 , of
m2 = 0.368m0 − 0.063m0 = 0.295m0 .
Second stage
After burn out,
mf = e−1 m2 = e−1 0.295m0 = 0.109m0 .
The fuel burnt in this stage will be m2 − mf = (0.295 − 0.109)m0 = 0.186m0 and the weight of the tank
plus engine will be 0.019m0 , leaving for the payload
m2stage
payload = (0.109 − 0.019)m0 = 0.090m0 ,
which is still very small but about twice the size of the payload obtained for the single stage rocket.
Note
Launch Strategy
The earth rotates towards the East with a period of about 23.9 hours. This results in a tangential velocity
of vt = 466m/sec. at the equator. The corresponding value for higher latitudes is vt = 466m/sec ∗ Cosθ0 ,
where θ0 is the latitude of position on the earth. In most situations, we desire to use this tangential velocity
to add to the energy available to place a satellite in orbit. Therefore, most launch sites are located as close
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to the equator as possible and the launch occurs with a velocity component directed towards the east. The
table shows the latitude of the current major international launch sites. Of interest is ”Sea Launch”, which
leaves the US west coast with a sea-going launch platform and control ship and launches very close to the
equator. Other criterion for launch site is safety; a failed launch should not put populations in danger.
28.5o
Kennedy
o
Tanegashima
30.4
Vandenburg
34.4
Xichang
28.25
Baikonur
45.6o
Kourou
5.2
Sea Launch
0
0 19 min
Note
Gravity Turn
The model we have introduced for rocket launch is incomplete if our goal is to place a spacecraft into orbit.
Orbital launch requires that the flight end with a roughly horizontal velocity at orbital speed. One useful
maneuver to accomplish this transition is called the gravity turn. In this maneuver, gravity acts to turn the
trajectory of the rocket towards the horizontal.
This maneuver offers two main advantages over a conventional thrust-controlled trajectory where the rocket’s
own thrust is used to steer the vehicle. First, any thrust used in changing the rocket’s direction is not being
used to accelerate the vehicle into orbit, constituting a loss which can be reduced by using gravity to steer
the vehicle onto its desired trajectory. Second, and more importantly, because the force of gravity is doing
the steering during the initial ascent phase of the launch the ship can maintain low or even zero angle of
attack. This minimizes transverse stress on the launch vehicle; allowing for a weaker, and thus lighter,
launch vehicle. For a constant thrust, and a variable mass, the details of the trajectory and motion require
a numerical solution. Using local intrinsic coordinate, the governing equations for the gravity turn are
v̇ = T /m(t) − g cos β
(19)
vβ̇ = g sin β
(20)
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Note
Specific Impulse
The specific impulse Isp , with units of seconds, is often used in practice to characterize the performance of a
rocket engine. The definition of the specific impulse is the magnitude of the thrust divided by the propellant
weight flow rate,
Isp =
T
c
= .
mg
˙
g
Typical values of Isp are around 300 s for solid propellants and up to 500 s for higher energy fuels.
ADDITIONAL READING
J.L. Meriam and L.G. Kraige, Engineering Mechanics, DYNAMICS, 5th Edition
4/6, 4/7
References
[1] M. Martinez-Sanchez, Unified Engineering Notes, Course 95-96.
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16.07 Dynamics
Fall 2009
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